Question: Find an ordered pair $(x,y)$ that satisfies both of the equations below: \begin{align*} 2x - 3y &= -5,\\ 5x - 2y &= 4. \end{align*}
Multiplying the first equation by 5 and the second equation by $-2$ gives
\begin{align*}
10x-15y&=-25,\\
-10x + 4y &=-8.\\
\end{align*}Adding the two equations gives $-11y = -33$, so $y=3$.  Substituting $y=3$ in the first original equation gives $2x - 9 = -5$, so $2x = 4$ and $x = 2$.  Therefore, the solution is $(x,y) = \boxed{(2,3)}$.